Questions tagged [cauchy-product]

For questions about the Cauchy product, the discrete convolution of two sequences.

118 questions
Filter by
Sorted by
Tagged with
41 views

What is the Laurent series of $e^{z+z^{-1}}$? (Detailed derivation using Cauchy product of series)

We know that: $$e^{z+z^{-1}}=\underbrace{\left(\sum_{n=0}^{\infty}\frac{z^n}{n!}\right)\cdot\left(\sum_{m=0}^{\infty}\frac{z^{-m}}{m!}\right)}_{=:\alpha}.$$ Now I am having trouble, applying the ...
58 views

50 views

86 views

An example: a convergence series, a divergent series, whose Cauchy product is convergent.

How to find an example: a convergence series $\sum a_n$, a divergent series $\sum b_n$, whose Cauchy product $\sum c_n$ with $c_n=\sum_{i+j=n}a_ib_j$ is convergent? Is there a simple example?
76 views

The Cauchy Product : Calculate the coefficients, $c_n$, in the product.

I found this definition to be different from other ones and the answer makes me feel weird. If $$A_{N}(x)=\sum_{n=0}^{n=N}a_{n}x^n$$ and $$B_{N}(x)=\sum_{n=0}^{n=N}b_{n}x^n,$$ calculate the ...
172 views

Integrating the square of an infinite series

Just out of plain curiosity, I want to know how to evaluate the integral of the square of an infinite series. For example, if $$f\left(x\right)=\sum_{n=0}^{\infty}c_n\left(x-a\right)^{n},$$ where $c_n$...
238 views

Are the unconditionally convergent series, with terms in a Banach algebra, closed under the Cauchy product?

We have a Banach algebra $\mathbb L$, and two sequences $(A_0,A_1,A_2,\cdots),\;(B_0,B_1,B_2,\cdots)\in\mathbb L^{\mathbb N}$, for which the sums $\sum_{n\in\mathbb N}A_n,\;\sum_{n\in\mathbb N}B_n$ ...
303 views

The necessity of absolute convergence in the convergence of the Cauchy product of series?

The Mertens' theorem claims that Suppose $\sum_{n=0}^\infty a_n,\sum_{n=0}^\infty b_n$ are two convergent series of complex numbers, convergent to $A,\beta$ respectively. If $\sum_na_n$ converges ...
178 views

44 views

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$ I believe that I have to use the cauchy product? But how do transform the expression to be a product ...
210 views

Product of Two Summations with Different Upper Limits

I'm trying to multiply two different finite summations with different upper limits. I've tried Cauchy Product but i think it's valid for same upper limits. I also tried to split the summation. Any ...
58 views

Find the power series of $f(x)=\frac{x}{x+1}$ in $x_0=0$

In 1st Semester Calculus book I found an exercise that asks me to find the above power series of the function at the point $x_0 = 0$ using the geometric series formula and the Cauchy-Product. So far I'...
212 views

69 views

Cauchy product of power series with $x^{2n}$

I am trying to rewrite a function $y(x)=\frac{1}{1+x+x^2+x^3}$ as a series. I used geometric series and got to two power series that I need to Cauchy product, however, one of them has $x^{2n}$ and I ...
Cauchy product for the reciprocal of the polynomial $x - 2x^2 + 3x^3 - 4x^4$
I have come across some Laurent series in which the denominator of a fraction contains a power series. Looking around I came across this Calculate Laurent series for $1/ \sin(z)$ which suggests that ...